Math / Trigonometry

QuestionIs $\cos \frac{5 \pi}{4}$ positive or negative? positive negative B

Studdy Solution

STEP 1

1. The cosine function is periodic with a period of $2\pi$ .
2. The angle $\frac{5\pi}{4}$ is in radians.
3. The cosine function is negative in the third quadrant of the unit circle.

STEP 2

1. Identify the quadrant of the angle $\frac{5\pi}{4}$ .
2. Determine the sign of the cosine function in that quadrant.

STEP 3

Convert the angle $\frac{5\pi}{4}$ into degrees to identify its quadrant. The conversion factor from radians to degrees is $\frac{180}{\pi}$ .
$\frac{5\pi}{4} \times \frac{180}{\pi} = \frac{5 \times 180}{4} = 225^\circ$

STEP 4

Determine the quadrant of $225^\circ$ . The quadrants are divided as follows: - First Quadrant: $0^\circ$ to $90^\circ$ - Second Quadrant: $90^\circ$ to $180^\circ$ - Third Quadrant: $180^\circ$ to $270^\circ$ - Fourth Quadrant: $270^\circ$ to $360^\circ$
Since $225^\circ$ falls between $180^\circ$ and $270^\circ$ , it is in the third quadrant.

STEP 5

Determine the sign of the cosine function in the third quadrant. In the unit circle: - Cosine is positive in the first and fourth quadrants. - Cosine is negative in the second and third quadrants.
Since $\frac{5\pi}{4}$ is in the third quadrant, $\cos \frac{5\pi}{4}$ is negative.
The cosine of $\frac{5\pi}{4}$ is $\boxed{\text{negative}}$ .

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